Why does pouring Coke or any other carbonated beverage on car battery
terminal corrosion cause it to dissolve and easily wash away?  I would
like a breakdown of the chemical reaction that occurs in such a case -
it doesn't make sense to me why mixing an acidic substance with an
acidic substance would cause this reaction.

H(+) + Fe2O3------------->H2O + Fe(3+)

Fe2O3 is alkali

I believe kava is on the right track. The acid (H+ ion) in coke would
dissolve rust, and most other oxide corrosion products.

The terminals themselves shouldn't be acidic, unless you have a battery leak.

It has been many a year since I studied any chemistry, but I don't
think there is any true reaction that takes place at all. It doesn't
evolve into a new compound...you just end up with a mixture. Coke and
crud. The tv show "MythBusters" tested the urban legend in question
and found that Coke seemed to be no more effective than any other
liquid at removing the battery corrosion.

There are a number of anecdotal uses of Coke as a cleaning product.
Perhaps why it works to some degree is more because of its foaming
action that anything else.  Hydrogen Peroxide might be as good or
better battery terminal cleaner, and it should be an oxidant!
I'm an ASE certified mechanic, and have a article on battery stuff on
my website: http://econofix.com/battery.html The success of coke or
anything else is probably due to the final step: a good rinse with
water, then drying before reassembly. Water and a good battery
terminal cleaner tool are all I ever use: I save my Coke for break
time!

Urban myth. I have experimented with this and shown it not to be the
case. Interestingly coke IS good for chrome cleaning with a bit of
tinfoil.

The only active ingredient of interest here would be the carbonic acid I think.

Coke and many other soft drinks contain high levels of carbonic acid,
which dissolves many substances easily, while still being weak enough
to leave others unharmed. Carbonic acid eats away limestone and most
metallic oxides quite readily, while not dissolving pure metals as
quickly.